[MichaelBurge]

Sure, that's the story the government tells everyone. It sounds plausible enough, but is it the whole story?

* Apply Downward Lowenheim-Skolem to get a countable model of set theory

* Construct the reals using whichever technique you want.

* Since the base set theory is countable, so is the set of constructed reals.

The same technique can give you countably many groups, countably many rings, countably many points in the plane, etc. Everything is countable.

[openasocket]

You're referring to https://en.wikipedia.org/wiki/Skolem%27s_paradox ?

It's a rather limited edge-case in mathematics, it's misleading to simply assert that "everything is countable."

[chriswarbo]

Your theory will have countable models, but via Goedel (and assuming we're ignoring inconsistent theories) it will be incomplete, capturing only a subset of the reals' properties.

Any particular theory, capturing some aspect of the reals, can be given a countable model. Those "constructed reals" are countable, but they miss out almost all of the reals.

Think about it this way: a theory is like an interface, a theory of the reals provides an API to access the reals. The API is necessarily limited; not least because there are only countably many ways we can combine the operations.

Now consider a mock implementation of that interface: rather than passing around reals, we invent some (countable) dummy objects to use instead. Since the API is limited, we can always come up with a mock implementation which cannot be distinguished from the real implementation using that API/theory.

That doesn't mean that the reals are countable. It does mean that uncountability is an implementation convenience; regardless of what result we're after, we could get it using countable objects, but we might have to spend a lot of effort "out smarting" the algorithms we're using (i.e. coming up with our "mock" set).

[danharaj]

Countable in the external logic, but the internal logic still proves that there are uncountably many reals.

[wayn3]

it is very much not possible to construct the real numbers in such a way that they are countable. (the set of real numbers is the object that "happens" when you fill the "holes" in the set of rational numbers).

cantors diagonalization argument is proof of that. you can't pull some silly trick to make them countable. there are many properties of R that are countable, but that doesn't make R itself countable.

[btilly]

The truth of your statement depends on what philosophy of mathematics you accept. Which itself is not something that can ever be settled by pure reason.

Here is a definition of the reals to consider. A real number is a computer program which implements a function f from positive integers N to the rationals such that |f(n) - f(m)| < 1/n + 1/m. This is a reasonable definition of real numbers, and all real numbers that you're likely to hear of are real numbers under this definition.

But now consider. There are a finite number of symbols that we build programs out of. Therefore for every N, there are a finite number of possible computer programs of length N. Only some of which represent real numbers under this definition. Therefore there is a countable sequence which contains all real numbers in it!

What is the key philosophical difference between this system of mathematics and the usual one? Quite simply this. Classical mathematics asserts the existence of things that cannot be written down or calculated by humans. This system of mathematics denies the existence of things which we cannot write down.

Now about a hundred years ago there was a major debate between these two philosophical camps. In the end the classical school won simply because most mathematicians don't care about philosophy, and classical mathematics is easier to work with. But the purportedly inescapable logical conclusions that you're taught about are not actually as inescapable as you've been lead to believe.

[powera]

I haven't the slightest idea what "n" and "m" are supposed to be, so I can't make any sense of your definition.

Beyond that, you say: "all real numbers that you're likely to hear of" - this is a fallacy twice over.

First, since my lifetime is finite, it's easy to count all the real numbers that I will hear in my life. But I don't think that's the argument you want to make.

Second, there are people who professionally construct unlikely real numbers; just because you choose to ignore those numbers doesn't mean that your countable sequence of all real numbers can. There's a difference between "things that cannot be written down" and "things you can't write down".

[btilly]

The statement "you are likely to hear of" is a concession that I know of numbers which can't fit that definition. Their key characteristic being that they cannot be calculated by humans, because evaluating them requires operations that are not actually possible for us.

Whether or not such numbers are well-defined at all is a point of philosophy. What does it mean to declare the truth of statements that cannot be verified either way?

[aaachilless]

I don't think I buy your argument, but I'm absolutely not a mathematician, so I could have missed something. Why I don't buy it:

It seems you didn't need to redefine the reals as programs to make this argument--the reals are classically defined as Cauchy sequences of rationals and (if the argument were valid) you could make the same claim using these sequences instead of programs.

I don't think the actual philosophical question is whether the reals are countable, it's whether the reals "exist". You can't define the reals such that they're countable because if you did, they wouldn't be the same structure.

[btilly]

Here is what you have missed.

The difference between the classical version and the constructive version that I wrote down is in what kind of rules you can use to construct a Cauchy sequence. My version is something that can be written down in a Turing machine. The classical version is that rules are anything that can follow a set of axiomatic rules - and includes rules that we cannot, even in principle, think of trying to evaluate.

Therefore in the classical version, you can create a rule that depends on being able to evaluate every other rule - including rules similar to itself. That is exactly what Cantor's argument does. In the constructive version you can only create rules using operations that we can guarantee will finish. Which doesn't include verifying the incalculable truth or falsity of arbitrary other rules.

Moving on, though, existence is critical. The key philosophical difference that drove the debate was pure existence proofs. Is proof by contradiction a valid method of reasoning with infinite sets? In particular, can you establish the existence of something by proving a contradiction if it does not exist? In what sense does the thing proven to exist actually exist? What if we prove that it exists but have no way to find it? What it we prove that it exists, have no way to find it, and no way to verify that we have found it?

These are not hypothetical questions. See the https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo... for a class of graph theory problems, all of which have polynomial time algorithms to solve. But we have no way to find those solutions. And in some cases it is impossible for us to verify whether a purported solution is a solution even if we were handed it.

So..do you believe in the existence of things in an infinite set that are impossible to find and impossible to verify?

[MichaelBurge]

And then there's the Axiom of Choice and all that it brings. There's a Hamel Basis for the reals which is provably impossible to construct(since it's equivalent to AC).

Do you believe in the existence of things that are provably impossible to construct?

[btilly]

As it happens, I don't.

That said I'm quite able to quote and do mathematics that depends on axioms that I do not accept. I can be quite the formalist when it is convenient. But I privately think of it as formal bullshit. And don't like seeing people who haven't accepted it beaten over the head until they do.

[Smaug123]

For that matter, do you believe in really, really big integers?

[openasocket]

You can't just define the reals to be something, you have to show that your construction is identical to the other constructions of the reals, like Dedekind cuts or Cauchy sequences. And yours doesn't: you can't represent Chaitin's constant in your definition, which is a real number.

[btilly]

If you wish to take that approach, you have to show that the other constructions of the reals actually construct something that exist. Which you can't. Just like you can't prove that ZFC is consistent.

That's why this is a philosophical question that is foundational for mathematics.

[openasocket]

> If you wish to take that approach, you have to show that the other constructions of the reals actually construct something that exist

First you said the reals are countable, now you're saying they don't exist at all?

What does it mean for something to "exist"? I can construct these sets from the axioms of ZFC, and under ZFC I can show your proof doesn't work.

Philosophy only enters into it when you are considering which axioms to take.

[btilly]

Philosophy only enters into it when you are considering which axioms to take.

I agree that by the time you start assuming ZFC, you're well past the realm of philosophy. The flip side of it is that in a discussion of philosophy you shouldn't make assertions based on axioms that have not yet been agreed on.

In any constructivist axiom system, the usual constructions of the real numbers do not define anything sensible. If you try to make sense of Cauchy sequences from a constructivist point of view, then you'll necessarily wind up with a definition that is very much like the one that I gave.

So you get the result that I stated. From a constructivist point of view, the usual "constructions" of the real numbers are nonsense and construct nothing, while the definition that I gave constructs something that can be reasonably called the real numbers. According to classical mathematics, both sets of constructions are well-defined, and the one that I gave is both complicated and different than the usual reals.

Which version you accept depends on your philosophical beliefs. Seeing what a different philosophical belief will lead to is very hard the first time you do it. But if you believe that it makes no sense to talk about the "truth" of unverifiable statements, you won't wind up debating the finer points of whether to accept C in ZFC...

[wayn3]

tl;dr: Your argument doesn't make any sense. Don't copy math arguments from hipster blogs or wikipedia. They rarely make any sense.

I am fully aware of the constructionist approach to mathematics.

But your argument goes somewhat like this:

The reals are real

In computer programs however, we can only do so much

Every computer program represents a real number

For every bounded amount of bits, there is only a countable set of computer programs

Therefore, the reals are countable.

This is not how math works. You extrapolate from a bounded set that is countable to the collection of all those bounded sets - and then claim because every one set is bounded, the collection must be bounded as well - and therefore countable.

Let me make a similar argument just to show how silly your argument is. Here we go: 1. Every set of integers with only one element is finite. Call it a trivial set, for example: {1}

2. Every finite union of such trivial sets is finite

3. Every finite union of such a finite union of trivial sets is finite

4. and so on - after many iterations, your finite set will look pretty big to the human eye - almost as if it was the set of integers.

5. Therefore, the set of integers must be finite.

Sounds pretty silly, right? Thats what you've done for a different property of a different set.

That is nonsense. This is why arguing with illiterates about math is so annoying. They think they can write non-stringent arguments and then claim they have "somewhat proof". In math, there is no such thing as "somewhat proof". You are either right or you are wrong. The reals are the reals by definition. You can not make them countable.

What you can make countable is sets of numbers that you work with. Sure. You wont ever create a computer that has access to all the reals, PRECISELY because it operates from a position of countable-ness (even quantum computers). But thats not what I care about when I talk math.

The real numbers are not accessible to in a "reality" way. They lie fully in the realm of arcane mathematics. Like many other things do. You would never claim that the hyper-exponentiation operator had anything to do with reality, and it doesn't. Neither do the reals. You're just confused about them because they appear so prominently in middle school mathematics.

Accept it and move on.

Mathematics is the realm of absolute truth. Everything we have proven is de facto true. You can make an argument that we are doing the "wrong kind of mathematics". And that may actually be true. But it doesn't change anything about our discipline. If you want to get philosophical and create a new Mathematics discipline based on different arbitrary rules (axioms), you are free to do so. But you will have to prove a lot of very mundane things and go through centuries of early day fuckups and its very likely that you will soon seek to merge with the already established mathematics.

Call it math2.0. I'll play with it. If its interesting, why not. But don't expect your new math to be more interesting than the one we've got.

[btilly]

You are quick to dismiss the argument in every way possible, including ad hominem attacks, but you failed to actually try to comprehend it. Just as you failed to try to verify your ad hominem attack about my mathematical ignorance. (Here is a hint. People whose expertise comes from quoting "hipster blogs and wikipedia" usually can't turn around and point to things like http://dspace.library.uvic.ca:8080/bitstream/handle/1828/277... as evidence that they actually know some math.)

But let's move on to the critical point. You said this:

The real numbers are not accessible to in a "reality" way. They lie fully in the realm of arcane mathematics. Like many other things do.

You are lecturing me on the nature of the nature of existence of things which cannot be described or verified by any process that can be carried out or imagined by any process that can exist in our universe. In a discussion about whether we should accept any existence for such things. And are failing to address the point that assuming axioms that lead to the conclusion of existence is an entirely unprovable assumption!

Do you see the obvious problem here?